TOMASZ POLACIK Second Order Propositional Operators Over Cantor Space Abstract. We consider propositional operators defined by propositional quantification in intuitionistic logic. More specifically, we investigate the propositional operators of the form A* : p ~ 3q(p -- A(q)) where A(q) is one of the following formulae: (-~q ~ q) v -~-~q, (-~--q --* q) --* (-,--q V--q), ((--'~q --* q) --~ ('~-~q V-~q)) ~ ((-~-~q ~ q)v-~-,q). The equivalence of A*(p) to -~-~p is proved over the standard topological interpretation of intuitionistic second order propositional logic over Cantor space. We relate topological interpretations of second order intuitionistic propositional logic over Cantor space with the interpretation of propositional quantifiers (as the strongest and weakest interpolant in Heyting calculus) suggested by A. Pitts. One of the merits of Pitts' interpretation is shown to be valid for the interpretation over Cantor space. We consider second order intuitionistic propositional logic. It is obvious that all (second order) propositional formulae define propositional operators. In this paper we focus on the operators of the form A* :p ~-* qq(p - A(q)) for some monadic formulae in {-~, V, A, -+). The operator * defined by *(p) = 3q(p - -~-~qV-~q) has been investigated in [1] and [3]. It has been shown (cf [3]) that the equivalence ,(p) _-__ -~-~p holds over Cantor space and the reals but it does not hold over the interval [0, 1] with respect to the standard topological interpretation of second order intuitionistic propositional logic. It means, in particular, that • and some other operators defined by propositional quantification are not generally de- finable in {-~, V, A, ~) relative to topological models. In [2] A.Pitts modelled the propositional quantification in Heyting calcu- lus interpreting VpA and 3pA as the strongest and weakest interpolant for the formula A not containing the variable p. One can ask whether this interpre- tation of propositional quantifiers agrees with topological interpretations of second order intuitionistic propositional logic over certain topological spaces. It turns out that in the general case the answer is "no" because Pitts' in- terpretation does not coincide with the topological interpretation over [0, 1]. It is so since *(p) is equivalent -- according to Pitts' interpretation -- to -~-~p. This "negative result" seems to be isolated as *(p) is equivalent to -~p over Cantor space and the reals. We also prove that Pitts' interpreta- tion and the interpretation over [0, 1] coincide for other formulae of the form Presented by Jan Zygmunt; Received June 19, 1992; Revised November 5, 1992 Studia Logica 53: 93-105, •994. © 1994 KluwerAcademic Publishers. Printed in the Netherlands.